The Informal Logic of Mathematical Proof

Informal logic is a method of argument analysis which is complementary to that of formal logic, providing for the pragmatic treatment of features of argumentation which cannot be reduced to logical form. The central claim of this paper is that a more nuanced understanding of mathematical proof and discovery may be achieved by paying attention to the aspects of mathematical argumentation which can be captured by informal, rather than formal, logic. Two accounts of argumentation are considered: the pioneering work of Stephen Toulmin [The uses of argument, Cambridge University Press, 1958] and the more recent studies of Douglas Walton, [e.g. The new dialectic: Conversational contexts of argument, University of Toronto Press, 1998]. The focus of both of these approaches has largely been restricted to natural language argumentation. However, Walton's method in particular provides a fruitful analysis of mathematical proof. He offers a contextual account of argumentational strategies, distinguishing a variety of different types of dialogue in which arguments may occur. This analysis represents many different fallacious or otherwise illicit arguments as the deployment of strategies which are sometimes admissible in contexts in which they are inadmissible. I argue that mathematical proofs are deployed in a greater variety of types of dialogue than has commonly been assumed. I proceed to show that many of the important philosophical and pedagogical problems of mathematical proof arise from a failure to make explicit the type of dialogue in which the proof is introduced.Comment: 14 pages, 1 figure, 3 tables. Forthcoming in Perspectives on Mathematical Practices: Proceedings of the Brussels PMP2002 Conference (Logic, Epistemology and the Unity of the Sciences Series), J. P. Van Bendegem & B. Van Kerkhove, edd. (Dordrecht: Kluwer, 2004

Enthymemes and Topoi in Dialogue

In Enthymemes and Topoi in Dialogue, Ellen Breitholtz presents a novel and precise account of reasoning from an interactional perspective. The account draws on the concepts of enthymemes and topoi, originating in Aristotelian rhetoric and dialectic, and integrates these in a formal dialogue semantic account using TTR, a type theory with records. Argumentation analysis and formal approaches to reasoning often focus the logical validity of arguments on inferences made in discourse from a god’s-eye perspective. In contrast, Breitholtz’s account emphasises the individual perspectives of interlocutors and the function and acceptability of their reasoning in context. This provides an analysis of interactions where interlocutors have access to different topoi and therefore make different inferences. Readership: All interested in the pragmatics-rhetoric interface and in theories of meaning and coherence in dialogue and discourse

A Normative Pragmatic Perspective on Appealing to Emotions in Argumentation

This is the author's accepted manuscript. The original publication is available at www.springerlink.com.Is appealing to emotions in argumentation ever legitimate and, if so, what is the best way to analyze and evaluate such appeals? After overviewing a normative pragmatic perspective on appealing to emotions in argumentation, I present answers to these questions from pragma-dialectical, informal logical, and rhetorical perspectives, and note positions shared and supplemented by a normative pragmatic perspective. A normative pragmatic perspective holds that appealing to emotions in argumentation may be relevant and non-manipulative; and that emotional appeals may be analyzed as strategies that create pragmatic reasons and assessed by the standard of formal propriety or reasonability under the circumstances. I illustrate the explanatory power of the perspective by analyzing and evaluating some argumentation from Frederick Douglass's "What to the Slave is the Fourth of July." I conclude that a normative pragmatic perspective offers a more complete account of appealing to emotions in argumentation than a pragma-dialectial, informal logical, or rhetorical perspective alone, identifies a range of norms available to arguers, and explains why appealing to emotions may be legitimate in particular cases of argumentation

A probabilistic analysis of argument cogency

This paper offers a probabilistic treatment of the conditions for argument cogency as endorsed in informal logic: acceptability, relevance, and sufficiency. Treating a natural language argument as a reason-claim-complex, our analysis identifies content features of defeasible argument on which the RSA conditions depend, namely: change in the commitment to the reason, the reason’s sensitivity and selectivity to the claim, one’s prior commitment to the claim, and the contextually determined thresholds of acceptability for reasons and for claims. Results contrast with, and may indeed serve to correct, the informal understanding and applications of the RSA criteria concerning their conceptual dependence, their function as update-thresholds, and their status as obligatory rather than permissive norms, but also show how these formal and informal normative approachs can in fact align

Evidence, Proofs, and Derivations

The traditional view of evidence in mathematics is that evidence is just proof and proof is just derivation. There are good reasons for thinking that this view should be rejected: it misrepresents both historical and current mathematical practice. Nonetheless, evidence, proof, and derivation are closely intertwined. This paper seeks to tease these concepts apart. It emphasizes the role of argumentation as a context shared by evidence, proofs, and derivations. The utility of argumentation theory, in general, and argumentation schemes, in particular, as a methodology for the study of mathematical practice is thereby demonstrated. Argumentation schemes represent an almost untapped resource for mathematics education. Notably, they provide a consistent treatment of rigorous and non-rigorous argumentation, thereby working to exhibit the continuity of reasoning in mathematics with reasoning in other areas. Moreover, since argumentation schemes are a comparatively mature methodology, there is a substantial body of existing work to draw upon, including some increasingly sophisticated software tools. Such tools have significant potential for the analysis and evaluation of mathematical argumentation. The first four sections of the paper address the relationships of evidence to proof, proof to derivation, argument to proof, and argument to evidence, respectively. The final section directly addresses some of the educational implications of an argumentation scheme account of mathematical reasoning

Pushing the bounds of rationality: Argumentation and extended cognition

One of the central tasks of a theory of argumentation is to supply a theory of appraisal: a set of standards and norms according to which argumentation, and the reasoning involved in it, is properly evaluated. In their most general form, these can be understood as rational norms, where the core idea of rationality is that we rightly respond to reasons by according the credence we attach to our doxastic and conversational commitments with the probative strength of the reasons we have for them. Certain kinds of rational failings are so because they are manifestly illogical – for example, maintaining overtly contradictory commitments, violating deductive closure by refusing to accept the logical consequences of one’s present commitments, or failing to track basing relations by not updating one’s commitments in view of new, defeating information. Yet, according to the internal and empirical critiques, logic and probability theory fail to supply a fit set of norms for human reasoning and argument. Particularly, theories of bounded rationality have put pressure on argumentation theory to lower the normative standards of rationality for reasoners and arguers on the grounds that we are bounded, finite, and fallible agents incapable of meeting idealized standards. This paper explores the idea that argumentation, as a set of practices, together with the procedures and technologies of argumentation theory, is able to extend cognition such that we are better able to meet these idealized logical standards, thereby extending our responsibilities to adhere to idealized rational norms